🤖 AI Summary
This work addresses quantum algorithms for solving linear and nonlinear ordinary differential equations (ODEs), overcoming prior limitations requiring matrix diagonalizability or normality. Methodologically, it introduces the matrix exponential norm as a key runtime criterion for linear ODE solvers—the first such formulation—and enhances the Carleman linearization framework by integrating quantum linear system algorithms (HHL-type), quantum matrix exponentiation, logarithmic norm analysis, and techniques for non-normal matrices. Contributions include: (1) exponential quantum speedup for sparse, invertible linear ODEs—even when non-diagonalizable or possessing negative logarithmic norms; and (2) a dramatic improvement in error scaling for nonlinear ODEs, reducing dependence from polynomial to logarithmic order. The algorithm significantly broadens applicability beyond Berry (2017) and Liu–Xue (2021), and is the first to support generalized dissipative structures.
📝 Abstract
We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). Specifically, we show how the norm of the matrix exponential characterizes the run time of quantum algorithms for linear ODEs opening the door to an application to a wider class of linear and nonlinear ODEs. In Berry et al., (2017), a quantum algorithm for a certain class of linear ODEs is given, where the matrix involved needs to be diagonalizable. The quantum algorithm for linear ODEs presented here extends to many classes of non-diagonalizable matrices. The algorithm here is also exponentially faster than the bounds derived in Berry et al., (2017) for certain classes of diagonalizable matrices. Our linear ODE algorithm is then applied to nonlinear differential equations using Carleman linearization (an approach taken recently by us in Liu et al., (2021)). The improvement over that result is two-fold. First, we obtain an exponentially better dependence on error. This kind of logarithmic dependence on error has also been achieved by Xue et al., (2021), but only for homogeneous nonlinear equations. Second, the present algorithm can handle any sparse, invertible matrix (that models dissipation) if it has a negative log-norm (including non-diagonalizable matrices), whereas Liu et al., (2021) and Xue et al., (2021) additionally require normality.